I'm using Visual Studio and getting meaningless errors from the linker. Double clicking on them takes me to lines in the .ino, but the errors themselves seem truncated to me. 

Is this the case?

Version numbers are:
Visual Studio Community Edition 12.0.3
Visual Micro 1503.18
Arduino 1.6.1
Teensyduino 1.21

This hardware and software configuration can happily compile, link, and upload other projects.

The output text is:
...
Hat_VS_VM_1_6_1.cpp.o:In function `Display::~Display()'
Hat_VS_VM_1_6_1.cpp.o:In function `setup'
Hat_VS_VM_1_6_1.ino:TaskScheduler(TimedTask**, unsigned char)'
Hat_VS_VM_1_6_1.ino:setShouldRun(bool)'
Hat_VS_VM_1_6_1.ino:setShouldRun(bool)'
Hat_VS_VM_1_6_1.ino:setShouldRun(bool)'
Hat_VS_VM_1_6_1.ino:setShouldRun(bool)'
Hat_VS_VM_1_6_1.ino:setShouldRun(bool)'
M_1_6_1\teensy31\Hat_VS_VM_1_6_1.cpp:55: more undefined references to `TimedTask::setShouldRun(bool)' follow
Hat_VS_VM_1_6_1.cpp.o:In function `setup'
Hat_VS_VM_1_6_1.ino:run()'
Hat_VS_VM_1_6_1.cpp.o:In function `loop'
Hat_VS_VM_1_6_1.cpp.o:In function `__static_initialization_and_destruction_0'
Hat_VS_VM_1_6_1.h:BVSensor(unsigned long)'
Hat_VS_VM_1_6_1.h:ClockTick(unsigned long)'
Hat_VS_VM_1_6_1.h:TemperatureSensor(unsigned long)'
Hat_VS_VM_1_6_1.h:StatusLED(unsigned char, StatusLEDMode, unsigned short)'
Hat_VS_VM_1_6_1.h:SwitchModes(unsigned long)'
Hat_VS_VM_1_6_1.h:RotarySwitch(unsigned long)'
Hat_VS_VM_1_6_1.h:TestHarness(unsigned long)'
Hat_VS_VM_1_6_1.h:MotionSensor(unsigned long)'
Hat_VS_VM_1_6_1.h:PatternGenerator(unsigned long)'
Hat_VS_VM_1_6_1.h:PatternGenerator(unsigned long)'
Hat_VS_VM_1_6_1.h:Display(unsigned long)'
collect2.exe*:error: ld returned 1 exit status
Error creating .elf

That's a worry. TimedTask is defined in a user library called HappyLEDs. This includes the class definitions in a set of files, one per class, to keep things tidy. These are all included in a utility sub-folder in the HappyLEDs folder.

So Hat_VS_VM_1_6_1.h has:
#include <HappyLEDs\HappyLEDs.h>

And HappyLEDs.h includes the files for each class with:
#include "utility/TimedTask.h"
#include "utility/TaskScheduler.h"
#include "utility/SwitchModes.h"
#include "utility/BVSensor.h"
etc...

That's weird...

It wasn't compiling with the Arduino IDE (both 1.0.5 and 1.6.1). I expected that, due to Visual Micro needing the libname prefix. However, removing that results in the error:
In file included from Hat_VS_VM_1_6_1.ino:2:0:
Hat_VS_VM_1_6_1.h:14:23: fatal error: HappyLEDs.h: No such file or directory

In the header file, I've tried:
#include <HappyLEDs\HappyLEDs.h>
#include "HappyLEDs\HappyLEDs.h"
#include <HappyLEDs.h>
#include "HappyLEDs.h"

All give the same error.

I say this is weird, because the HappyLEDs.h file is in the right place (....\workspace\libraries\HappyLEDs), the IDE is looking in the right place, the file has a valid name (matches the directory name and doesn't have prohibited characters), and the content is valid. If I use the Import Library functionality in both the Arduino IDE and Visual Studio, the HappyLEDs library appears in the list of libraries and the code inserted by that functionality is #include <HappyLEDs.h>.

Further fiddling around reveals that the error occurs if the library include is in the main header file (Hat_VS_VM_1_6_1.h) whereas if the #include <HappyLEDs.h> line goes at the start of the Hat_VS_VM_1_6_1.ino file, then there is no error and my code compiles successfully in both the Arduino IDE and Visual Studio/Visual Micro.

I remain somewhat confused by the Arduino build process and the resulting errors. In this case, putting the library includes in the .ino works and putting them in the .h doesn't.